On the exceptional set in the generalized binary Goldbach problem
type: Item
instance of: scholarly article
title: On the exceptional set in the generalized binary Goldbach problem
zbMATH Open document ID: 0961.11033
author: G. I. Arkhipov
author: K. Buriev
author: V. N. Chubarikov
published in: Doklady Mathematics
publication date: 2001-01-28T00:00:00Z
review text: Let \(\beta\) be irrational, and for \(x\to+\infty\) let \(T_1(x)\) denote the cardinality of the set of positive integers which are not representable as \(p+[\beta q]\), where \(p\) and \(q\) are primes, and \([\cdot]\) denotes the integral part. The authors prove that \(T_1(x)\ll_{\varepsilon} x^{1-2/9+\varepsilon}\) (though somewhat stronger bounds are true for suitable values of \(\beta\), depending on the structure of its continued fraction). Furthermore, the authors study a similar problem for the sum \(p+[q^c]\), where \(c>1\) is not integral, and obtain the corresponding bound \(T_2(x)\ll_{\varepsilon} x^{1-2\delta+\varepsilon}\) for the exceptional set, where \(\delta\) depends on the saving over the trivial bound in a suitable exponential sum. Actually this paper contains only the statement of the main results and of the most important lemmas (mainly estimates for exponential sums), without any reference to the complete proof.
reviewed by: Alessandro Zaccagnini
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 1557640
zbMATH Keywords: Goldbach problem
zbMATH Keywords: binary problems with primes
zbMATH Keywords: exponential sums
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1557640